Question: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{k^3 - 9k^2 + 14k}{10k^2 - 60k - 70}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {k(k^2 - 9k + 14)} {10(k^2 - 6k - 7)} $ $ a = \dfrac{k}{10} \cdot \dfrac{k^2 - 9k + 14}{k^2 - 6k - 7} $ Next factor the numerator and denominator. $ a = \dfrac{k}{10} \cdot \dfrac{(k - 7)(k - 2)}{(k - 7)(k + 1)}$ Assuming $k \neq 7$ , we can cancel the $k - 7$ $ a = \dfrac{k}{10} \cdot \dfrac{k - 2}{k + 1}$ Therefore: $ a = \dfrac{ k(k - 2)}{ 10(k + 1)}$, $k \neq 7$